2.3 Numerical tests
2.3.3 Second study case: 9 diffusivity zones equation
We consider the same problem as before, but subdividing Ω in 9 sub-domains. So we consider Ω = ∪9i=1Ωi such that
Ω1= [0, 1/3] × [0, 1/3], Ω2 = [1/3, 2/3] × [0, 1/3], Ω3 = [2/3, 1] × [0, 1/3], Ω4 = [0, 1/3] × [1/3, 2/3], Ω5 = [1/3, 2/3] × [1/3, 2/3], Ω6 = [2/3, 1] × [1/3, 2/3],
Ω7= [0, 1/3] × [2/3, 1], Ω8 = [1/3, 2/3] × [2/3, 1], Ω9 = [2/3, 1] × [2/3, 1], and equation (1.11) with g ≡ 1 and b of the form
b(x; y) =
9
X
i=1
yi1Ωi(x), for x ∈ Ω,
for some fixed y = (y1, . . . , y9) ∈ (0, ∞)9. As before, we consider the stochastic problem: find u : Γ → H01(Ω) such that, a.e.,
a(u(Y ), v; Y ) = f (v) ∀ v ∈ H01(Ω), (2.20) where a(·, ·; y) denotes the bilinear form (1.10) with b = b(·; y) and Y = (Y1, . . . , Y9) is a random vector taking values in Γ =Q9
i=1Γi =Q9
i=1[ai, bi]. Analogously, we suppose that Γi= [1, 3] and Yi are i.i.d. random numbers such that
Yi− 1
2 ∼ Beta(αi, βi), (2.21)
for i = 1, . . . , 9. Again we implemented diverse versions of weighted greedy and POD algorithms for the construction of reduced order basis spaces, taking Vδ as the classical P1-FE approximation space. We hence compared them, confronting the values of error (2.15), as a function of the reduced order basis space dimension N .
As one can observe, the only difference with the problem of section (2.3.2) is practically just the higher number of parameters. We made two different tests for different parameter values: the first one with αi = βi = 10, for i = 1, . . . , 9, the second one with αi = βi = 75, for i = 1, . . . , 9.
0 1 2 3 4 5 6 7 8 9 N
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
log10(E[∥uNδ(Y)−uN(Y)∥
2 H1 0
(Ω)])
error
comparison
Standard POD Monte-Carlo POD Gauss-Jacobi POD
Figure 2.5: Comparison of error (2.14) obtained using standard, Monte-Carlo and Gauss-Jacobi POD algorithms with, respectively, M = 100, 100, 256.
0 1 2 3 4 5 6 7 8 9
N
−6.0
−5.5
−5.0
−4.5
−4.0
−3.5
−3.0
−2.5
−2.0
log10(E[∥uNδ(Y)−uN(Y)∥
2 H1 0
(Ω)])
error
comparison
Standard Greedy Weighted Greedy Standard POD Gauss-Jacobi POD
Figure 2.6: Comparison of error (2.14) obtained using standard Greedy and weighted greedy and standard and Gauss-Jacobi POD algorithms with, respectively, M = 1000, 1000, 100, 256.
Figure 2.7: Geometrical set-up of problem (2.20).
Greedy algorithm
We implemented the standard and the weighted greedy algorithm for construction of reduced basis space, taking ω = √
ρ in the weighted case as before. Again, for the train set selection, we sampled it using various techniques. We took
|Ξ| = 2000 and we chose the first parameter µ1 as µi1 = 2, for i = 1, . . . , 9 (2 is the mode of the distribution of Y ). The best accuracy is still achieved using the weighted algorithm with a sampling of the distribution of Y . In Figure 2.8, we reported the graph of the error (2.14) (in a logarithmic scale) as a function of the reduced basis space dimension N , using a standard greedy algorithm (with an uniform sampling) and weighted greedy algorithms with a uniform sampling or a sampling of Y for αi = βi = 10 (left) and αi = βi = 75 (right). In the first case we observe that the weighted algorithm with sampling of Y performs better than the standard one. However, the weighted algorithm with uniform sampling does not show a much better accuracy, providing an accuracy hardly different from that of the standard POD. This implies that with an higher dimension of the parameter space the sampling technique assumes much more importance. This is strongly underlined in the case αi = βi = 75. Here, the distribution of Y is much more concentrated and the difference between the standard greedy and the weighted one with sampling of Y is more evident. Moreover, the performance of the weighted algorithm with uniform sampling gets much worst than the standard one: this is clearly due the fact that, having a bad sampling, the weighted greedy algorithm forces us to take points µ1, . . . , µN for which the respective solutions are almost linearly dependent. This is highlighted by the fact that for N ≥ 8 the reduced matrix generated by the greedy algorithm becomes singular.
0 2 4 6 8 10 12 14
error comparison: firs parame er S andard Greedy Weigh ed Greedy - Uniform Weighted Greedy - Distribution
Figure 2.8: Comparison for the error (2.14) obtained for αi = βi = 10 (left) and αi = βi = 75 (right), using standard greedy algorithm and weighted greedy algorithms with uniform sampling or sampling of Y.
POD method
We implemented the various weighted versions of POD algorithm and compared the results obtained in the cases αi = βi = 10 and αi = βi = 75. In Table 2.2 we reported the weights and the number of sample points used in the various algorithms. Since Gauss-Legendre and Gauss-Jacobi POD algorithms are based
wi |Ξ|
Standard 1 500
Uniform Monte-Carlo ρi 2000
Monte-Carlo 1 500
Gauss-Legendre ωiρi 512
Gauss-Jacobi ωi 512
Table 2.2: Description of the weights used in the weighted POD algorithms and of the number of points of Ξin the two different trials.
on a tensor product rule the only possibility was to take M = 29 = 512. Indeed the next possible choice is M = 39 = 19683, which is computationally impracticable.
We did not tested Clenshaw-Curtis POD, since |Ξ ∩ ˚Γ| = 0 for M = 29 and
|Ξ ∩ ˚Γ| = 1 for M = 39, so that for having an enough representative set Ξ we would need M ≥ 49, which is clearly impracticable. Figure 2.9 shows that the weighted algorithms perform better of the standard one, even if Monte-Carlo POD outperforms uniform Monte-Carlo POD. The situation is highlighted in the case αi = βi = 75, where the distribution of Y is much more concentrated. Moreover, for obtaining good results for uniform Monte-Carlo POD we had to take M = 2000 while we run Monte-Carlo POD with M = 500 (for bigger M the performance does not change). The difference between the two weighted algorithms can be addressed to the fact that rule (2.17) is a better approximation of (2.14) than (2.16) and to the more representative choice of the points in Ξ. Therefore, it seems that, in the case of high dimensional parameter space, the choice of the sampling points plays
a key role in the POD algorithm, as in the greedy. In Figure 2.10 we reported
error comparison: first parameter Standard POD
error comparison: second parame er S andard POD Mon e-Carlo POD Uniform Monte-Carlo POD
Figure 2.9: Comparison for the error (2.14) obtained for αi = βi = 10 (left) and αi = βi= 75 (right), using standard, uniform Monte-Carlo and Monte-Carlo POD.
the performances of Gauss-Legendre and Gauss-Jacobi POD algorithms. As in the case considered in section 2.3.2 Gauss-Jacobi POD shows performances almost equal to the Monte-Carlo POD ones. However this time Gauss-Legendre performs very poorly. This is due to a bad selection of the sampling points Ξ by the Gauss-Legendre algorithm: from N ≥ 8, for both parameter values, the accuracy shows hardly any change for higher N . Finally, in Figure 2.11 we confronted the two
0 2 4 6 8 10 12 14
error comparison: first parameter Gauss-Legendre POD
error com arison: second arameter Gauss-Legendre POD Gauss-Jacobi(75,75) POD
Figure 2.10: Comparison for the error (2.14) obtained for αi = βi = 10 (left) and αi = βi = 75 (right), using Gauss-Legendre and Gauss-Jacobi (of same parameters of distribution of Y ) POD.
well working weighted POD algorithms (Monte-Carlo and Gauss-Jacobi) with the standard one. In both the cases a better accuracy is obtained with the weighted algorithms. The difference is much more visible in the case αi = βi = 75, when the distribution is highly concentrated.
Greedy vs POD: a comparison
We report in Figure 2.12 the comparisons of the errors obtained using standard and weighted reduced order approximation. We selected, both for POD and greedy algorithms, the weighted versions with the best performance. We observe
0 2 4 6 8 10 12 14
error comparison: first parameter Standard POD
error compariso : seco d parameter Sta dard POD Mo te-Carlo POD Gauss-Jacobi(75,75) POD
Figure 2.11: Comparison for the error (2.14) obtained for αi = βi = 10 (left) and αi = βi= 75 (right), using standard, Monte-Carlo and Gauss-Jacobi (of same parameters of distribution of Y ) POD.
that in both cases a better accuracy is obtained with a weighted algorithm instead of a standard one. The differences between weighted and standard algorithms are much more evident when the distribution is highly concentrated.
0 2 4 6 8 10 12 14
error comparison: first parameter Standard Greedy
error comparison: second parameter Standard Greedy Weighted Greedy Standard POD Monte-Carlo POD
Figure 2.12: Comparison of error (2.14) obtained using standard Greedy and weighted Greedy and standard and Monte-Carlo POD algorithms for αi = βi = 10 (left) and αi = βi= 75 (right).